Method and device for obtaining tridimensional optical image formation from bidimensional measurements of attenuation of radiation through an object

ABSTRACT

The use of the derivative of the Radon transform is used for obtaining three dimensional images or reconstructions of the examined objects. The derivative provides a precise reconstruction of the image as opposed to the use of Radon transforms themselves. The device is particularly suited for three dimensional imaging and x-ray apparatus.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a method for obtaining tridimensionaloptical image formation of an object by means of irradiations, as wellas a device used for the application of this method. A tridimensionalreconstruction is effected by the processing of a series ofbidimensional measurements of attentuation of radiation through theobject, the incidence of radiation being modified between saidmeasurements.

2. Description of the Prior Art

Bidimensional optical image formation from bidimensional measurements ofthe attenuation of radiation is well-known. The equipment comprises asource of suitable radiations, such as X-rays for medical examinations;the object to be examined is placed between the irradiating source and asheet of paper or sensitive film whose points have an image producedaccording to the intensity of the rays at the outlet of the object; thecontrasts observed concerning the image indicate the position of theabsorbent zones of the object

However, the information thus obtained is inadequate for certainapplications and consequently methods for the tridimensionalreconstruction of the object have been proposed.

Magnetic resonance tomography methods require the use of costlyinstallations and extremely wide uniformity of the magnetic field wherethe object to be examined is placed. Moreover, the measuring timesrequired to embody a tridimensional reconstruction are extremely long.Consequently, these drawbacks limit the advantages of these methods.

It has even been proposed to establish tridimensional reconstructions bythe superimposition of bidimensional reconstructions or sections of theobject. A collimation of an X-ray source makes it possible to obtain afan-shaped radiation which traverses one section of the object and thenproduces an image of a line of sensors; the source rotates around theobject so as to irradiate the same section under different angles; thesuccessive measurements are stored and a computer makes it possible todetermine local contribution on attenuation in each point of the meshingof the section. The source and the sensors are then offset and carriedonto another section around which they move according to a trajectoryparallel to the previous one.

The examination time thus depends on the adopted number of trajectories.In practice, it is unfortunately not possible to allow for an axialsampling as compacted as the sampling inside a section; the object alsorisks moving between examinations of the two sections, which increaseslocalization uncertainties.

A further drawback is linked to collimation, which reduces the energyefficiency of the source and which may require that the device bestopped from time to time during the examination so as to allow it tocool down.

Methods have also been proposed which use a conical beam which rotatesaround the object and by which it is possible to carry out severalirradiations which provide many bidirectional images of the object. Ifthere is a sufficient number of these images, a computer can analyze andcombine these images in order to reconstitute a tridimensional image ofthe object. These methods use what is known as the transformed Radon ofthe attenuation of radiation at each point of the meshing of the object.The transformed Radon of a function at a point is equal to all the sumsof the local values of this function on each plane passing through atleast one point of the range where the function is delivered. Inpractice, this clearly satisfies a discrete topology with a finitenumber of planes in order to describe the transformed Radon and a finitenumber of points in order to describe the function.

Measurement of attentuation of radiation on a line of sensors of a flatdetector placed behind the object provides attenuation according to abeam of rays contained in a plane of the Radon space.

The sum of the attenuation along this line gives the value of thetransformed Radon of attenuation for this plane. The numerical inversionof the transformed Radon gives attenuation at all points of thedefinition range of the function. It needs to be acknowledged that thisentire process is complicated and that certain of the methods proposedresult in erroneous results being obtained or all results neverthelessbeing inaccurate.

Amongst the literature available, reference may be made here to thearticle by Schlindwein and entitled "Iterative three--dimensionalreconstruction from twin--cone beam projection" (IEEE Transactions onnuclear science, vol. NS-25, n°4, October 1978, pages 1135-1143), Wherethe methods using the transformed Radon are rejected owing to theircomplexity, and that of Minerbo entitled "Convolutional reconstructionfrom cone--beam projection data" (IEEE Transactions on nuclear science,vol. NS-26, n°2, April 1979, pages 2682-2684) which uses a methodimplementing the transformed Radon.

As revealed subsequently by the invention, the use of the transformedRadon itself nevertheless requires the use of approximations in thenumerical calculations, and moreover the articles of the prior Art donot provide concrete devices allowing for reconstructions of goodquality of tridimensional images.

SUMMARY OF THE INVENTION

The invention can overcome these drawbacks. First of all, it concernstridimensional optical image formation devices, all of which comprise asingle conical radiation source in front of the object and a detectorbidimensional beam behind the object, the source and network beingmobile along diverse incidences with respect to the object. It alsoconcerns a method implementing the first derivative of the transformedRadon of attenuation of the radiation on the points of the object, itscalculation and its inversion.

The samplings, required to obtain acceptable results, are mentioned. Adetailed flowchart is proposed which in particular explains theinterpolations.

A further object of the invention is to furnish trajectories of thesource and the device for measurements to be made around the object andwhich are compatible with the method.

The invention first of all concerns a tridimensional optical imageformation device by the irradiations of an object and comprising aradiation source irradiating a conical-shaped space in which the objectis placed, a detector comprising a bidirectional device measuringattentuation of the radiation having traversed the object, a mechanismmaking it possible to carry out a series of irradiations of the objectunder different influences, as well as a measurement chain and acomputer which analyses and processes the information of thebidimensional device during irradiations so as to deduce from this thelocal contribution at the different points of a meshing representing theobject on attenuation of the radiation, wherein the source is unique andthe computer comprises units suitable for carrying out the calculationand completing inversion of the derivative of the transformed Radon of afunction being defined as all the local values of this functionconcerning each plane passing through at least one point of the rangewhere the function is defined, and the derivative of the transformedRadon being defined as the sum of the variation rates concerning each ofsaid planes if movement occurs perpendicular to said plane in thedirection of the normal vector defined by a system of sphericalcoordinates.

According to one possible embodiment, the mechanism comprises a circularrail centered on an origin and on which the source carries out theirradiations according to a circular trajectory, the detector movingonto this same trajectory and occupying opposing positions with respectto the origin.

According to a more elaborate embodiment, the mechanism comprises twoparallel circular rails, two sections on which slide respectively arethe source and the detector both placed with opposing positions withrespect to the origin. The sections pass through the circulartrajectories when the irradiations are carried out. Moreover, these arebent back in the form of an arc of a circle in such a way that thesource and detector remain at a constant distance from the origin.

The invention also concerns a method for the tridimensional opticalimage formation of an object from bidimensional measurements of theattenuation of a radiation through the object by the use of a deviceformed of a conical radiation source comprising a focal spot and abidimensional detector formed of a network of sensors, wherein, in eachof the first points of a meshing representing the object, attenuation iscalculated of the radiation by calculating the variables representativeof the derivative of the transformed Radon of attenuation of radiationof the points of a second meshing associated with the transformed Radonof the object, the transformed Radon of a function being defined asbeing all the local values of this function concerning each planepassing through at least one point of the range where the function isdefined, and the derivative of the transformed Radon being defined asthe sum of the variation rates concerning each of said planes ifmovement occurs perpendicular to said plane in the direction of thenormal vector defined by a system of spherical coordinates, thequantities being calculated by adding up for each second point thevariation of attenuation of radiation along at least one line obtainedby intersection of the detector with a plane passing through the focalspot of the conical radiation and in the proximity of the second point,the straight line passing through the origin and the second point beingroughly orthogonal to the plane passing through the focal spot, thenlinear combinations of these addings up, and the attentuation ofradiation in each of the first points being obtained by derivation ofthese quantities with respect to the original distance, and finally bylinear combination of the derived quantities, interpolations beingeffected moreover in order to pass from the second points to the firstpoints.

The method may be advantageously used in the case where the source andthe detector pass through two trajectories at a constant distance fromthe origin roughly in the form of a sinusoid comprising at least twoperiods out of a complete revolution around the object and whoseamplitude is equal to or greater than the distance between the originand any point of the object, the distance between the points of thetrajectories and the origin being moreover sufficient so that any planepassing through the object will encounter the trajectory. For twoperiods, this is verified if this distance is equal to or greater thanthis amplitude multiplied by √3.

The method for reconstructing images according to the invention involvesa special calculation method. For the calculation of a parameterconcerning the first points of a first tridimensional meshing of thepoints of the object whose cartesian coordinates are generally evenlydistributed, this firstly consists of defining second and third pointswhich constitute a second and third tridimensional meshing ofcharacteristic points, the spherical coordinates of the second pointsbeing evenly distributed and the second points belonging in particularto the meridian planes converging on an axis, the third points, whosecylindrical coordinates are evenly distributed, belonging both to themeridian planes and parallel planes containing the first points andorthogonal to the axis; then of obtaining information concerning thesecond points and of calculating derived information; then of combiningthis information concerning groups of second points belonging to thesame meridian planes so as to deduce from this intermediate informationconcerning the third points; and finally of combining the intermediateinformation concerning groups of third points belonging to the sameparallel planes so as to deduce from this the parameter concerning thefirst points.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention may be more readily understood by reading the followingannexed figures, which are in no way restrictive:

FIG. 1 shows the main parts of a device according to the invention, aswell as the notations used for explaining the method;

FIG. 2 shows a possible second device according to the invention;

FIG. 3 shows a possible third device according to the invention;

FIG. 4 represents a geometric construction which illustrates one stageof the method according to the invention;

FIG. 5 shows a geometrical construction which illustrates aninterpolation stage required for application of the method;

FIG. 6 essentially represents an advantageous double trajectory forconducting analyses by means of the device of FIG. 3;

FIG. 7 shows a synoptic diagram of the installation which pilots thedevices according to the invention;

FIGS. 8 and 9 represent two other devices according to the invention;and

FIG. 10 shows a flowchart of the method used.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

The device according to the invention comprises (refer to FIG. 1) asource 10 emitting a divergent conical beam whose focal spot is S-shapedand which traverses with attenuation an object 11 to be analyzed and forwhich it is desired to reconstruct the local contributions uponattenuation. The nature of the radiation may be any, such as X-rays,provided it is able to characterize an element of the object 11 forwhich it is desired to diagnose the presence or concentration of saidelement by an attenuation different from the one carried out by itsother elements.

An individual ray Ri traverses the object 11 between the input pointsMei and output points Msi; in particular, it traverses any supposedintermediate point M and which will serve as a base for the radiationsto which it will be exposed. It finally reaches a detector 12 andaffects a particular sensor 13 from amongst those of a bidimensionalnetwork on a screen 14 situated at a point K. It is marked by thecoordinates (p,q) of its track A on a detection plane passing through anorigin 0 orthogonal to OS and referenced Pdet on FIG. 1: this immaterialplane is introduced so as to allow for a simpler explanation of thereconstruction method without it being necessary to take account of theremoteness of the screen 14 and its eventual curve.

If the local contribution at the point M upon attenuation of theradiation is noted f(M), the sensor 13 thus measures a radiationintensity: ##EQU1## attentuation being supposed negligible outside theobject 11 and IO being the known and fixed intensity of the radiation Riwhich would have been measured in the absence of the object 11.

IO may also be supplied by a monitoring method evaluating the directflux on sensors 13 of the bidimensional network or the flux localizedoutside the measurement field in contact with the object 11 at theoutput of the source 10.

After logarithmic conversion, a measurement is thus made of attenuationof the radiation through the object along the ray Ri derived from S andpassing through A (stage 101 of FIG. 10): ##EQU2##

Sampling coefficients, calculated in an initial phase duringmeasurements concerning the objects whose attenuation is known, make itpossible to correct the data supplied by the sensors.

A bidimensional image 15 is obtained of the object 11 on the screen 14.Conventional radiography is formed by such an image. In order to obtaintridimensional reconstruction, it merely needs to print a rotatingmovement of the focal spot S of the source 10 around the object 11, forexample on a circle Ce centered on the origin O around the object 11 soas to irradiate it under the different incidences and then of combiningbetween them the representations obtained. It is clearly necessary thatthe detector 12 follows the movement of combining between them therepresentations obtained. It is clearly necessary that the detector 12follows the movement of the beam. Here, it is supposed that it alsomoves on the circle Ce. It may also move in another way, for example onanother circle centered on the origin O, but of a different diameter;only the increase in size of the image 15 is different. The trajectoryalong the circle Ce may be realized by hooking the source 10 anddetector 12 to a circular rail 20 whose diameter may be equal ordifferent from that of the circle Ce according to the hooking modeadopted.

However, one major problem appears: as the volume of information to bequickly stored is quite considerable, numerical resolution of thereconstruction problem would need to resort to using complicated matrixcalculations. However, the invention does not need to resort to thistype of method.

First of all, the transformed Radon from attenuation f(M) of theradiation is defined for those planes passing through a point M (Rf(M))which defines all the sums of the function describing the localcontribution on attenuation of the radiation at their points on each ofthese planes; these planes are each characterized by a pairing (OM.n, n)where N is a unitary vector perpendicular to the plane in question andOM.n is the algebraic measurement associated with the distance of theplane to the origin O. In order to identify these planes, acharacteristic point C is used, an orthogonal projection of the origin Oon the plane:

    OC=(OM.n)n.

Next, the first derivative of the transformed Radon of attentuation f(M)of the radiation is defined for those planes passing through a point M(R'f(M)) which is equal by definition for each plane P (OM.n, n) to thesum concerning all the points of the plane of the derivative of thefunction describing the local contribution on attenuation with respectto the direction of the vector n.

Amongst all these planes, it is possible to distinguish and note via theabbreviation PM a Radon plane P IOM.n, n) associated with acharacteristic point CM. n is defined in a marking linked to the object11 by its longitude φ measured in the plane of the circle Ce and by itscolatitude θ measured with respect to the axis of the circle Ce so thatθ=τ/2 when n belongs to the plane of the circle Ce and θ=+O or +τ whenit is perpendicular to this plane.

In order to determine the values of the transformed Radon Rf(M) or ofits first derivative R'f(M) at the point M, it is necessary toespecially calculate the sum of the function describing the localcontribution on attenuation f(M) or its derivative concerning all thepoints of the Radon plane PM associated with this point M. This ispossible if the focal spot S of the source 10 itself belongs to theRadon plane PM, since one fraction of the radiation or its closeproximity does not then leave this Radon plane PM or its close proximityand thus may not undergo attenuation by points of the object to considerthis fraction in order to evaluate the sum of the function describingthe local contribution on attenuation or of its derivative concerningall the points of the object 11 belonging to the Radon plane PM (theambient air is virtually not absorbent. If the object of interest iscontained in an absorbent environment, the outer environment shall beregarded as a disturbing factor).

However, calculation of the transformed Radon Rf(M) may thus only bemade approximately. The Applicant showed that the first derivative ofthe transformed Radon R'f(M) may on the other hand be calculatedexactly, which results in improved quality of the tridimensional images.

In the case of a source 10, whose focal spot S rotates around the object11 along a circle Ce, FIG. 4 shows a point M whose Radon plane PM cutsthe circle Ce at two points JG and JD. The focal spot S of the source 10must therefore be placed at one of these points in order to make itpossible to calculate the sum of attenuation or its derivative on theRadon plane PM.

The first derivative of the transformed Radon R'f of the object itselfis defined as all the summations concerning the Radon planes PM passingthrough at least one point M of the object of the derivative along thenormal vector n of the function describing the local contribution onattenuation. The Radon plane PM is common to all the points of theobject 11 it contains and in particular to the characteristic point CM.Associated to this point here is the value of the transformed RadonRf(M) or its derivative R'f(M) on the plane PM.

The characteristic volume of the object refers to all the characteristicpoints CM associated with all the Radon planes PM passing through atleast one point M of the object. The reconstruction of thetridimensional image may be carried out when attenuation is availablefor all these planes.

However, the measurements only permit access to one value of R'f for theplanes PM encountering the trajectory passed through by the focal spotS. The characteristic volume of the measurements refers to all thecharacteristic points CM associated with the Radon planes passingthrough at least one point of the trajectory. The characteristic volumeof the object must as far as possible be included in the characteristicvolume of the measurements.

In the case of a spherical object centered on the origin O and with theradius Rob and the circular trajectory Ce, the characteristic volume ofthe object is the same sphere centered on O and with the radius Rob. Thecharacteristic volume of the measurements is a torus To shown on FIG. 4and obtained by the rotation of a circle contained in the plane passingthrough the focal spot S and the axis of the circle Ce targeting theaxis at the level of the origin O and diameter SO.

Thus, it can be observed that the characteristic volume of themeasurements does not make it possible to cover the entirecharacteristic volume of the object: there remains a characteristicshadow zone of the planes encountering the object, but not the circleCe. The calculation of R'f concerning this zone may only be carried outby interpolation.

Irrespective of the position of the object, there will still be a shadowzone linked to the planes passing through the object, but do notencounter the circle Ce. In order to fill up this shadow zone, it isnecessary to renounce the plain circular trajectory and select atrajectory so that any plane passing through at least one point of theobject encounters the trajectory.

This condition may be concretely satisfied if the circular trajectory Ceis produced by the accompanying movement of the source 10 and thedetector 12 along a circular rail 20 and if associated to (FIG. 2) saidrail 20 is a mechanism 21 for pivoting an angle ξ along an axis passingthrough the origin O thus enabling it to assume two positionsmaterialized by the circles Ce1 and Ce2. A blocking may be provided byany suitable means for these two positions. This can be easily realizedwhen the nearer the angle ξ is to τ/2, the more the maximum radius Robof the object sphere centered on O, whose characteristic volume does nothave any shadow zone, increases. For ξ being worth τ/2, that is for twoperpendicular trajectories, the maximum value for Rob is Rc/ √2 if Rcdenotes the radius of the circles Ce1 or Ce2.

FIG. 3 shows another possibility for embodying such a trajectory: thedevice here comprises two circular parallel rails 30 and 30' with thesame radius R30 each passed through by two diametrically opposedsupports 32 and 34 and 32' and 34' respectively. The supports 32 and 32'constitute the extremities of a section 31 along which the source 10slides by means of handles 36; similarly, the supports 34 and 34'constitute the extremities of a section 33 along which the detector 12slides by means of handles 35. The sections 31 and 33 are the arcs ofcircles centered on the origin O .

Thus, it is possible to combine two rotations for the source 10 and thedetector 12 in order to replace them concerning trajectories belongingto two concentric spheres. Subsequently, we suppose them to beidentical.

One possible example is the one where the focal spot S of the source 10passes through a trajectory Tc shown on FIG. 6 and with an equatione=Int.cos2ψ where ψ denotes an angle of rotation along the rails 30 and30', e a clearance parallel to the axis of the rails 30 and 30' with e=owhen the focal spot belongs to the plane passing through the origin Oand parallel to the rails 30 and 30' and Int is one amplitude. It isthen possible to show that the transformed state of Radon Rf(M) or itsderivative R'f(M) can be obtained for all the characteristic points CMof the object 11 if R1f≧Int.√3 and Int≧Rob where R1f is the radius ofthe smallest sphere centered on the origin O and containing all of theobject 11.

During displacement of the source 10, the point of attachment of thedetector 12 moves at the same time onto the points of a trajectory Tc'symmetrical with respect to the origin O.

The advantage of the device of FIG. 3 with respect to that of FIG. 2 isto allow for the carrying out in one continuous rotation of themeasurements at the time the double circular trajectory imposes a doublerotation and a stop time for pivoting and similarly for a givenprocession speed an examination time at least twice longer. Moreover,the circular double trajectory introduces significant redundancy in themeasurements since there is a high proportion of planes which encounterboth of the two trajectories. One alternative to the device of FIG. 3consists of providing the points 21 of FIG. 2 with a motor 22 controlledby a computer 50 (FIG. 7) by means of a line 23 so as to cause the angleξ to vary when the focal spot S moves onto the circle Ce. Any trajectorymay be obtained in the tridimensional space. Considering the respectivemasses of the radiation sources X and bidimensional detectors, such asthe luminosity amplifiers, the device of FIG. 3 seems preferable.

FIG. 3 also shows the device which controls the movement of the detector12 on the section 33. It comprises an electric motor 40 connected to acomputer 50 (FIG. 7) via a line 44 and whose shaft is ended by a gear 42which gears inside a rack side of the section 33; as the movementsrequire great accuracy, another side of the section 33 comprisesgraduations 43 marked by an optical sensor 41; it then sends a signal tothe computer 50 via a line 45 and stopping of the motor 40 is controlledby the line 44. An identical motor and optical sensor also control themovement of the source 10 on the section 31; on the other hand, thereare similar devices with an electric motor 140 and an optical sensor 141respectively connected to the computer 50 by lines 144 and 145 in orderto move the supports 32, 32' and 34, 34' along the rails 30, 30' whichthus also comprise a rack and graduations.

The devices of FIGS. 1 and 2 may all also be piloted by these devicesalthough these have not been represented here.

The movements of the source and the screen may generally be independentand synchronized; they may also be obtained by means of a single motorand a mechanical linking effected, for example, by a bar or, moregenerally, a rigid mechanical structure.

In order to examine the object 11, other devices are also possible.

Firstly, as shown on FIG. 8, it is possible to envisage a deviceinvolving a joint movement of the source 10 and the detector 12connected by a rigid mechanical structure 65 rotating under the actionof a motor 66 controlled by the computer 50 around the object 11. Thisstructure comprises a vertical column 67 which is also the output shaftof the motor 66 and from which project are two opposing radial arms 68and 69. The first arm 68 terminates by a first pole 70 to which thesource 10 is suspended, the second arm 69 by a second pole 71 to whichthe detector 12 is suspended. In this embodiment as in the previousembodiments, the object 11 is laid on a support transparent to radiationand shown here by 72.

As shown by FIG. 9, it is still possible to apply the invention to adevice in which the source 10 and the detector 12 are fixed and hookedto an immobile structure 75 and, on this figure, are shown respectivelyat the top and bottom of the object 11.

The object 11 is moved in rotation by means of a fork 76 whose sleeve 80rotates inside a bearing 77 provided inside the structure 75 around anaxis roughly perpendicular to the radiation emitted by the source 10.Two bearings, referenced respectively as 81 and 82, are provided insidethe two branches 78 and 79 of the fork 76 and receive two pivot pinssituated on both sides of a flat frame 83 which thus rotates between thetwo branches 78 and 79 around an axis perpendicular to the axis ofrotation of the fork 76.

The object Il is rigidly secured, for example by compression, betweentwo compression plates 84 at one

The object 11 is rigidly secured, for example by compression, betweentwo compression plates 84 at one extremity of telescopic rods 85, theother extremity being integral with the frame 83; the telescopic rods 85have a common directrix and spread towards each other and are pushed bysprings 86 around or inside the rods and are compressed between theframe 83 and the compression plate 84 respective to the frame 83. Thus,it is possible to submit it to any rotation and carry out the sameexaminations as with the other devices described up until now. Thecompression plates and possibly the rods are assumed to be transparentto radiation from the source 10.

The rotation movements of the fork 76 and the frame 83 are controlled bythe computer 50 by means of motors (not shown).

This device is not suitable for the examination of a human being, but isof interest for the nondestructive examination of small objects.However, it is necessary to limit the rotations so as to avoid radiationtraversing the frame 83 and the branches 78 and 79 or to also suitablyprovide these parts in a transparent material.

In order to describe the method for reconstruction of the image by meansof the derivative R'f of transformed Radon, new notations (figure I)need to be defined. Reference is made to the detection plane Pdetassociated with the position of the focal spot S of the source 10, theplane being perpendicular to the axis OS and passing through the originO. This plane is provided with a Cartesian mark (u,v) so that the mark##EQU3## is direct. The vector u is selected parallel to the plane ofthe circle Ce. This detection plane Pdet serves to define thecoordinates p and q of the sensors 13 on the detection screen 14. Thesummation straight line denoted as D(OM.n,n) or DM is the intersectionof the Radon plane P(OM.n,n) or PM associated with the point M with thedetection plane, and 0' being the point of DM, an orthogonal projectionof the point O on the straight line DM. The straight line DM isorientated by a unitary vector V1 so that: forms a direct mark. This iscalled the angle between the values v and v1.

It is also necessary to define the two weighted attenuation functions:##EQU4## where Rc is the radius of the trajectory on which the focalspot S moves from the source 10, namely the distance OS. These functionsare independent of the Radon plane in question.

Associated to them for each straight line DM are the functions SY(S,M)and SZ(S,M) respectively representing the entirety of the functions Yand Z on the summation straight line.

This shows, which is the start point of the numerical calculations,that: ##EQU5## where β represents the angle between the vectors n andOS.

The first formula (1) is an approximate relation linking themeasurements to the transformed Radon Rf. The second (2) is an exactrelation linking the measurements to the first derivative of thetransformed Radon R'f; it makes it possible to obtain more preciseresults and remains verified irrespective of the distance between thesource 10 and the object 11. It thus enables it to be brought togetherto the maximum, whilst the approximation of the formula (1) is that muchmore accurate when the source 10 is distanced from the object 11, whichimposes the use of an awkward-shaped device.

The formula (2) is mathematically equivalent if the following is posed:##EQU6## with the two formulae: ##EQU7## where A(p), respectively A(q)represent the point A of the summation straight line DM of absciss prespectively of ordinate q.

Preferably, the formula (5) is selected for α close to O modulo τ andthe formula (4) is selected for c close to τ/4 modulo τ.

As shown by the flowchart of FIG. 10, for a given position of the source10, the quantities X(S,A) and Y(S,A) are thus calculated of any point Aof the detection plane Pdet (stages 101 and 102). The derivatives ofY(S,A) with respect to the coordinates p and q of the detection planePdet are completed by the filtering operations (stage 103) by using theconvolution of the function Y(S,A) by two differentiation filters. Thesource 10 is then moved as far as its next position (stage 104) and thecycle recommences until all the acquisitions have been carried out.

The subsequent stages consist of carrying out calculations forcharacteristic points U' of the characteristic volume of the object.These calculations introduce the Radon plane PM which is associated withthem, as well as the intersection Radon straight line DM of the Radonplane PM with the detection plane Pdet. Calculation starts as follows(stage 105):

in the case of formula (4): ##EQU8## and in the case of formula (5):##EQU9## by summation concerning the Radon straight lines DM and then alinear combination (stage 106) allowing the following to be obtained:

in the case of formula (4): ##EQU10## in the case of formula (5):##EQU11## after which these quantities are standardized (stage 107) bymultiplying them by: ##EQU12## then by 1/|sin α| or 1/|cos α|respectively. The result is equal to R'f(U).

If the system of coordinates (p,q) is not adapted for fixed sampling bythe distribution of the sensors on the detector 12, these formulae maysubsequently be transposed into the appropriate system of coordinates.It is also possible inversely to recalculate by interpolating theattentuations into the system of coordinates (p,q), thus carrying outwhat is called a recorrection or resampling of the measurements.

In the case where a Radon plane encounters the trajectory at severalpoints, for example two (JG and JD) in the case of FIG. 4, it is usefulto carry out an average concerning all or at least one part of thevalues of R'f associated with each of these points so as to reduce thestatistical errors linked to noise concerning these measurements.Generally, an average is nevertheless selected relating to twopreferential points.

In the case where a Radon plane associated with a characteristic pointCM does not encounter the trajectory at any point, which occurs if thecharacteristic volume of the measurements leaves a shadow zone withinthe characteristic volume of the object, it is desirable that it benevertheless allocated a value R'f via an interpolation procedure. Inorder to do this, it is possible, for example, to set up aninterpolation of about zero: associated with the characteristic pointCM, outside the characteristic volume of the measurements, is thesurface corresponding to the intersection of the sphere centered on Oand passing onto CM with the characteristic volume of the measurements.Then on this surface, a point C'M (not shown) is selected at a minimumdistance from CM. By definition of the characteristic volume of themeasurements, the plane P'M (not shown) admitting as a characteristicpoint C'M shall encounter the trajectory at one or more points, most ofthe time remaining as a tangent to these. The preceding method makes itpossible to define a value R'f associated with C'M. The approximatelyzero interpolation consists of allocating this same value to the pointCM.

Thus, a value R'f is allocated to all the points of the characteristicvolume of the object. As already mentioned, it is of course preferableto avoid interpolation regarding the shadow zone and thus to use adevice, such as the one represented on FIG. 3. However, for moremechanical simplicity of the device, it is possible to be satisfied withthe simple circular trajectory Ce and tolerate the interpolation inorder to fix the values concerning the associated shadow zone.

It merely remains to deduce f(M) from knowledge of the first derivativeR'f of the transformed Radon concerning the object characteristicvolume. This inversion operation is straightforward and can beimplemented by high-capacity calculations.

For a given unitary vector n, the algebraic measurement (radius) P isnoted of the points U' concerning the origin axis O and directive vectorn'.

The theoretical inversion formula of the first derivative of thetransformed Radon R'f is written as follows: ##EQU13##

Note that ω and θ denote the longitude and colatitude of the point inquestion (here U') and ρ is defined by OU'=ρ.n.

However, it is necessary to look into the rendering discrete problemswhich have not been mentioned so far in this text.

In particular, it is evident that the source 10 may only carry outexposures under a finite number of determined incidences, that thedetection screen 14 carries out the measurements by means of a networkof sensors 13 marked on the detection plane as also a finite number, andthat the object 11 must be rendered discrete or also meshed.

Inevitably, interpolation problems arise. The recommended solutions andwhich also belong to the invention are described below. Simultaneously,an examination is made of the conditions relating to resolving therendering discrete problems so as to enable correct reconstructions tobe obtained.

The object 11, assumed to be entirely included in a sphere centered onthe origin O and with a radius Rob, may be described with the aid of thepoints M which result in rendering discrete an even parallelpipeddisplay or representation meshing whose coordinates verify:

    x(i)=[(2i-1-Nx)/Nx].Rob where 1≦i≦Nx,

    y(j)=[(2j-1-Ny)/Ny].Rob where 1≦j≦Ny,

    z(k)=[(2k-1-Nz)/Nz].Rob where 1≦k≦Nz.

If ν is the sought-after cut-off frequency for the imagery system 3D,Nx, Ny and Nz are preferably taken as being greater than or equal to4.ν.Rob.

The detection screen 14 is also advantageously graduated in Cartesiancoordinates, which moreover have already been introduced. The locationsof the sensors 13 defined by their coordinates on the detection planePdet verify:

    p(a)=[(2a-1-Np)/(Np-2)/(NP-2)].Rob where 1≦a≦Np,

    q(b)=[(2b-1-Nq)/(Nq-2)/(Nq-2)].Rob where 1≦b≦Nq.Z

Np and Nq are equal to or greater than 4.ν.Rob.

The points U of the sampling of the first derivative of the transformedRadon are defined with the aid of a spherical meshing (a patterndefining a volume in which points are arranged) centered on the originO; their spherical coordinates verify:

radius: p(n)=[(2n-1-Nn)/(Nn-2)].Rrad where 1≦n≦Nn, colatitude:θ(l)=[(2l-1)/2Nl].τ/2 where 1≦l≦Nl, longitude: φ(m)=[(m-1)/Nm].2τ where1≦m≦Nm. where Rrad is the radius of the sphere centered on Oencompassing the object characteristic volume. For an object sphere witha radius Rob, Rrad=Rob.

For Nn, an even number is preferably taken so as to not incorporate theorigin O in the meshing, as it characterizes an infinite number ofplanes. On the other hand, the following is chosen:

    Nm=2.Nn and Nl=Nn/2 for 2τ.ν.Rob≦Nm≦4τ.ν.Rob.

This makes it possible to reduce the artifacts linked to the inversionof the first derivative of the transformed Radon to an acceptable level.

The positions of the source 10 from which the attenuation measurementsare made are the intersections of the meridian planes orthogonal to themeridian planes of longitude (m) with the trajectory Ce or Tc.

The actual problem thus consists of calculating from the Nm positions,which may be assumed by the source 10, the first derivative of thetransformed Radon R'f(U) for the entire network of the points U and thenthe local contribution on attenuation f(M) for the entire network of thepoints M. This is arrived at with the aid of a series of interpolationstages. One possible method is given in detail in the continuation ofthis text.

For the points U forming part of the characteristic measurement volume,FIG. 5 shows that the derivative of the transformed Radon R'f(U) of anypoint U can be obtained for at least two generally different locationsS1 and S2 of the focal spot S for which the spheres, whose diameters arelimited by the origin O and respectively by the locations S1 and S2,meet at the point U. In practice, the locations S1 and S2 do not,however, correspond to the measurement positions of the focal spot S,but are each situated between two of these positions, namelyrespectively S11 and S12 and S21 and S22.

For the points U outside the characteristic measurement volume, thesource positions S1 and S2 are marked from the point of thecharacteristic measurement volume used to define the zero orderinterpolation referred to earlier.

In actual fact, the first derivative of the transformed Radon iscalculated for the points U11, U12, U21 and U22 forming part of all thepoints U' and being the nearest points of intersection of U of sphereswith respective diameters of OS11, OS12, OS21 and OS22 with the circlecorresponding to the parallel line of the point U (all the points withthe same radius (n) and the same colatitude θ(l)). If this intersectionis empty, the points of intersection circles are taken between thesespheres and the sphere centered on O and passing through the nearest U'sof the circle corresponding to the parallel line of the point U. Theirdistances of the point U are respectively d11, d12, d21 and d22.

The transformed Radon is calculated from the point U or its derivativevia the interpolation formula (7): ##EQU14## This operation correspondsto stage 108 of the flowchart.

It is clear that the computer which manages the process knows in advancethe position of the points S11, S12, S21, S22, U21, U12, U21, U22 forany point U of the sampling, as well as the distances d11, d12, d21 andd22 which are stored in the memory; the position of the summationstraight lines associated with the points U11, U12, U21 and U22 is alsoknown in advance and the attenuation values X(S,A) along its points areobtained by interpolation of the values measured on the sensors 13traversed by this straight line. In order to calculate the sumconcerning the points in question of any summation straight line, thecomputer 50 thus possesses weighting coefficients associated with eachsensor 13. With the proposed sampling, the points U number Nl×Nm×Nn=Nn³and there are 2.Nn positions of the source 10; for each of thesepositions, the transformed Radon or its derivative is calculated for Nn²summation straight lines DM. Then, an average is calculated concerningthe values combined two by two.

In order to carry out the numerical calculation of the formula (6), itis possible and advantageous to separate the integrals.

So as to describe the execution of this calculation, the notion of theplane of rearranged projections is introduced passing through the originO and whose points are at a constant longitude. Thus, this involvesmeridian planes.

To any point B of this plane and to any vector n with a longitude, thepoint CB is associated, an orthogonal projection of the point B on theaxis passing through the origin O and with a director vector n. When Bdescribes the plane of rearranged projections, the point CB describes aplane of characteristic points corresponding to the meridian planeassociated with the longitude φ. Geometrically, these two planes aremerged.

It may be noticed that if B is the orthogonal projection of the point Mon the plane of rearranged projections, the point CB is identical to thecharacteristic point CM associated with the plane DM defined by M and n.

It is thus possible to start by calculating concerning each plane ofrearranged projections from the values R'f relating to the associatedmeridian plane the following quantity: ##EQU15## for all the points Bwhich are orthogonal projections of at least one point M of the object.

Then secondly, the local calculation on attenuation is calculated foreach point M of the object: ##EQU16##

This method makes it possible to obtain a strict reconstruction of theobject 11 even if the characteristic volume of the object is entirelyincluded in the characteristic volume of measurements.

Moreover, the use of the exact formula makes it possible to bring nearerthe source 10 of the object 11. Thus, it is possible to reduce thespatial requirement of the device and increase the growth factor (focalspot S distance--detector 12/focal spot S distance--object 11) and thusimprove the spatial resolution of the device. Furthermore, thiscontributes in providing better use of the radiation to the extent thatthis enlarges the solid irradiation angle of the object. As theradiation sources are limited in their output of photons per angularunit, for a total number of photons required to pass through the objectduring the period of measurements, this enables the examination time tobe reduced and thus to increase temporal resolution. It also, byretaining the same examination time, allows statistical accuracy to beimproved as regards the reconstructed object.

The reconstruction diagram may be generalized to a wide class oftrajectories by authorizing, for example, that the distance of the focalspot from the source 10 or from the detector 12 is different, indeedeven variable. In the preceding formulae, the weighting coefficientslinked to growth shall then depend o the measurements.

It should be mentioned that the source 10 and the object 11 may also bebrought together if a trajectory is selected according to FIG. 6 andconforms to the formula:

    e=Int(cos.nψ)

the higher is n.

Inversion of the derivative of the transformed Radon, defined by theformulae mentioned above, is effected concretely with the aid of thepoints B of a meshing known as a rearrangement meshing plotted on theplanes of rearranged projections associated with the meridians of thepoints U by way of sampling:

    r(c)=[(2c-1-Nn)/(Nn-2].Rrad where 1≦c≦Nn,

    z(d)=[(2d-1-Nz)/Nz].Rob where 1≦d≦Nz,

    φ(m)=[(m-1)/Nm].2τ where 1≦m≦Nm,

where r(c) denotes the coordinate of a point B of a meridian plane alongthe axis parallel to the plane of the circle Ce, and z(d) the coordinateof a rearrangement point B along the axis of rotation of the circle Ce:on each meridian plane, the points B are evenly distributed on arectangular network.

Calculation of the quantity Q(ψ,B), such as the one defined earlier, isthus limited to the calculation of the quantities Q(ψ,B) for the pointsof the rearrangement meshing after an interpolation has enabled thederivatives δR'f(CB)/δρto be obtained from the derivatives δR'f(U)/δ'for each direction of the meridian (stage 111).

Calculation of the derivatives δR'f/δρ shall advantageously have beencarried out by digital processing techniques, as for example convolutionby the associated filters. The data would have also been firstlyweighted by the factor sinθ (weighting and filterings stages 109 and110).

According to known techniques, the filtering operations linked to thecalculation of δR'f/δρ and and the retroprojection operations linked tosummation concerning the colatitude θ may be replaced by oneretroprojection operation followed by one filtering operation withoutdeparting from the context of the present invention. The filteringoperation could also possibly be made on the object 11 after summationconcerning the angles θ and has been carried out.

The final interpolations to be carried out are made inside each plane ofthe object perpendicular to the meridian planes: one has seen that thecalculation of the local contribution on attenuation f(M) of the point Mintroduces the orthogonal projections of the point M concerning eachrearranged projection plane taken into account. These projections fallbetween the points B where it is necessary to interpolate the quantitiesQ(ω, B) before adding them up, thus realizing retroprojection operations(stage 112).

For these final operations, the system is also aware in advance of thepositions of the points of the different meshings and possesses, in theform of tables or matrixes, and possesses the coefficients to be takeninto account. The prior programming work is thus important and must berepeated if it is desired to have several different samplings, but thecalculations to be made during an examination remain reasonable andmainly comprise linear combinations and filterings.

A computer 50 manages the reconstruction process (FIG. 7). It consistsof a synchronization unit 51, a memory unit 52, a calculation unit 53and peripheral units 54. Initially, the synchronization unit 51 ensuresacquisition of the measurements: according to the indications of theoptical sensors 41 or 141, it activates the electric motors 40 or 140and stops them when the source 10 and the detector 12 are at apredetermined measurement position; then by means of a line 57, itindicates the start and end of the acquisitions on an ordinary chain ofmeasurements 55 connected by a line 56 to each of the sensors 13 whoseprocessed information is supplied by a line 58 to the memory unit 52.

When acquisition is terminated for a given position of the source 10,the synchronization unit 51 restarts the electric motors 40 or 140,whilst the sensors 13 are reset to zero. The next measurement may thenbe carried out and stored.

It is still possible to continuously move the source 10 and the detector12. The total time involved in executing the measurements is thenappreciably reduced as the motors are no longer stopped, but it isnecessary to accept a lack of focus of the image due to rotation duringeach irradiation.

It should be mentioned that the graduations 43, which indicate thepositions of the irradiations on the synchronization unit 51, are notnecessary; they can also be eliminated and the synchronization unit 51itself determines the cycle of irradiations by means of a clockincorporated in the synchronization unit.

After acquisition of all the measurements, the actual examination of theobject is completed. All the information passes into the calculationunit 53 via a line 59; the calculation of the transformed Radon or itsfirst derivative is made, as well as their inversion.

The values of the local contribution on attenuation f(M) are finallyrouted by a line 60 towards the peripheral units 54 which in particularcontain graph outputs and display screens.

The differentiation and interpolation operations described here must notbe merely considered as calculation tools and may be replaced by otheroperations without departing from the context of the present invention;similarly, the proposed samplings are merely examples and may varywithin relatively wide limits. Several of the operations of theflowchart may also be inverted.

However, it should be noted that the expression of the derivative of thetransformed Radon, the circular trajectory and the conical beam, foranalytic and numerical precision reasons, virtually impose working witha spherical beam of points U when it is natural to represent an object,even with a quasi-spherical shape, with the aid of a rectangular networkof points M in which flat sections may be defined. Several interpolationoperations are thus essential so as to pass from one network to theother. On the other hand, the density of the meshing of the points M ofthe object 11 does not strictly depend on the number of measurementsdefined by the number Nm: a certain flexibility is thus possible at thislevel.

It is finally clear that the devices involving movements of the source,object and the detector, said movements being different from thosedescribed here by way of illustration, still come within the context ofthe invention. Similarly, the measurements may be carried out morequickly by several sources functioning simultaneously, the views beingdistributed between these.

Accordingly, the invention provides an extremely useful method ofobtaining tridimensional reconstructions from bidimensional measurementsof the attenuation of a radiation. One plausible application isobviously medical imagery and a second is nondestructive control, butany object 11 compatible with the dimensions of the equipment and theradiations available could be examined. The algorithms used directlycalculate the solution, which accelerates processings comparedespecially with iterative methods. Finally, the method guarantees exactlocalization of the information measured: no distortion of the imageappears, which is not the case when an image is made by using othermethods of the prior Art.

According to the measurement device and the conventions used, differentstandardization factors or scale changes may be made (stage 113 of theflowchart). The local contributions on attenuation described in the textcorrespond to linear coefficients of attenuation. But it also possible,for example, to express the attenuation measurements as a length ofequivalent water, and then after reconstruction translate the result ofthe calculations by using the Hounsfield scale according to theconventions for medical scanners X.

I claim:
 1. A device for obtaining tridimensional images frombidimensional measurements of the attenuation of radiation through anobject comprising;a radiation source for irradiating a conical spacefrom a focal spot (S) in which the object is placed; a detectorcomprising a bidimensional device for measuring the attenuation of theradiation through the object, a mechanism for performing a series ofmeasurements under different incidences, as well as a chain ormeasurements; and a computer for the analysis and processing of theinformation of the bidimensional device at the time of the measurementsand for calculating from the local radiation contribution f(M) atdifferent points (M) of a meshing representing the object uponattenuation of radiation, wherein the source is unique and the computercomprises units for performing the calculation and inversion of thederivative of the transformed Radon of attention of radiation, amathematical function called a Radon transform of a function beingdefined as all the local values of this function concerning each planepassing through at least one point of the range where the function isdefined, and the derivative of the transformed Radon being defined asthe sum of the variation rates on each of said planes if movement occursperpendicular to said plane in the direction of the normal vectordefined by a system of spherical coordinates and wherein the derivativeof said transformed Radon is used for constructing an object.
 2. Aimagery device according to claim 1, wherein the source and detector arefixed at a constant distance from a fixed origin for the variousincidences.
 3. A device according to claim 2, wherein the mechanismcomprises a circular rail centered on the origin (O) for guiding thesource and the detector during the measurements so that they remain in afixed alignment with the origin (I) at a constant distance, with thesource's movement describing a circular trajectory Ce.
 4. A deviceaccording to claim 3, wherein the circular rail further comprises apivot for enabling the source to be moved according to two circulartrajectories (ce1, Ce2) offset from a constant angle (ξ),
 5. A deviceaccording to claim 3, wherein the circular rail further comprises apivot for controlling by a motor which is controlled by the computerenabling said motor to undergo a time-controlled variable rotation.
 6. Adevice according to claim 2, wherein the mechanism further comprise:twoparallel circular rails, made up of two sections on which sliderespectively the source and the detector said sections both being placedsuch that they are at opposing positions with respect to the origin(O)and such that said source and said detector transverse the circularrails so when a measurement is carried out, the sections moreover arebent back as an arc of a circle so that their points are at a constantdistance from the origin (O), such that the source and the detector canremain permanently aligned and at a constant distance from the origin(O).
 7. A device according to any one of claim 3 to 6, furthercomprising rials on which the source and the detector (12) move by meansof electric motors piloted by the computer with the aid of markingdevices.
 8. A device according to claim 6, further comprises means tomove said sections on said circular rails by means of electric motorspiloted by the computer (50) with the aid of marking devices.
 9. Adevice according to claim 2, wherein the source and the detector areconnected by a rigid mechanical structure.
 10. A device according toclaim 9, wherein the rigid mechanical structure pivots around an axispassing through the origin (O).
 11. A device according to claim 9,wherein the rigid mechanical structure is immobile and includes anothermechanical structure to which the object is secured and which enables itto carry out rotations according to at least one axis passing throughthe origin (O).
 12. A method for the tridimensional optical imageformation of an object for bidimensional measurements of the attenuationof a radiation through the object by using a device formed from aconical radiation source comprising a focal spot (S) and a bidimensionaldetector formed from a network of sensors (13), wherein in each of thefirst points (M() of a discretization representing the object, theattenuation of the radiation f(M) is calculated by calculatingquantities (R'f(U)) representative of the derivative of the transformedRadon of attentuation of the radiation of the points (U) of a secondmeshing of the object, the transformed Radon of a function being definedas all the local values of this function concerning each plane passingthrough at least one point of the range where the function is defined,and the derivative of the transformed Radon being defined as the sum ofthe variation rates concerning each of said planes if movement occursperpendicular to said plane in the direction of the normal vectordefined by a system of spherical coordinates, the quantities (R'f(U))being calculated by carrying out the summation for each second point (U)of the variation of attenuation of the radiation along at least one lineobtained by the intersection of the detector with a plane passingthrough the focal spot (S) of the conical radiation in the proximity fthe second point (U), the straight line passing through the focal spot(S), then linear combinations of these summations, and the attenuationsof the radiation at each of the first points (M) being obtained byderivation of these quantities with respect to the distance to theorigin (O), and by linear combination of the derived quantities,interpolations being moreover carried out so as to pass from the secondpoints (U) to the first points (M) such that an object can beconstructed.
 13. A tridimensional optical image formation methodaccording to claim 12, wherein the quantities (Ff(U)) are eachdetermined by summation of the variation of the attenuation along linesobtained by the intersection of the detector and planes each passingthrough an interpolation point (11, U12, U21, U22) close to thecorresponding second point (U), as well as through different positionsof the focal spot (S) of the conical radiation, the straight linedefined by the original (O) and each interpolation point (U11, U12, U21,U22) being orthogonal to the plane passing through said interpolationpoint.
 14. A tridimensional optical image formation method according toeither of claims 12 or 13, wherein the measurements are carried out whenthe focal spot (S) of the conical radiation occurs on meridian planesdistant from regular angles (2τ/Nm) and converging at an axis passingthrough the origin (O).
 15. A tridimensional Tridimensional opticalimage formation method according to claim 14, wherein the source and thedetector transverse two trajectories (Tc, Tc') at a constant distancefrom the origin (O) roughly in the shape of a sinusoid comprising atleast two periods over a complete revolution around the object.
 16. Atridimensional optical image formation method according to claim 14,wherein the source and the detector transverse two trajectories (Tc,Tc') at a constant distance from the origin (O), roughly in the form ofa sinusoid comprising two periods over a complete revolution of theobject and whose amplitude (Int) is equal to or greater than thedistance between the origin (O) and any point of the object (11), thedistance between the points of the trajectory (Tc) and the origin (O)being moreover equal to or greater than this amplitude multiplied by √3.17. A tridimensional optical image formation method according to claim16, wherein the second points (U) for which the quantities R'f(U) aredetermined have spherical coordinates (ρ(n), θ(1), φ(m)) evenlydistributed and belong to the planes perpendicular to the meridianplanes in which the focal spot (S) of the conical radiation occurs atthe time of irradiations.
 18. An optical image formation methodaccording to claim 17, wherein the derived quantities are weighted,interpolated and added up inside each meridian plane so as to obtainquantities Q (ρ, B) at third points (B) belonging to a third meshinghaving rectangular coordinates (r(a), z(b)) evenly distributed overplanes of rearranged projections of planes merged with the meridianplanes and belonging to the same planes orthogonal to the meridianplanes as the points (M) of the meshing representative of the object,and wherein also, for any point (B) of the third meshing, thecalculations of the quantity (Q, (θ,B)) characteristic of said point (B)are combined with those of the other points of the third meshingbelonging to the same plane of rearranged projections.
 19. Atridimensional optical image formation method according to claim 18,wherein the attenuation f(M) at any point (M) of the representativemeshing is obtained by linear combination of the quantities (Q (ρ, B))associated with the third points (B) of orthogonal projections of thepoints of the points (M) on the planes with rearranged projections. 20.A method for calculating a parameter (f(M)) on the first points (M) of afirst tridimensional meshing of an object whose Cartesian coordinatesare generally evenly distributed, wherein it consists firstly ofdefining second points (U) and third points (B) which constitute secondand third tridimensional meshings of characteristic points, thespherical coordinates of the second points (U) being evenly distributedand the second points (U) belonging in particular to meridian planesconverging at an axis, the third points (B), whose cylindricalcoordinates are evenly distributed belong to both the meridian planesand parallel planes containing the first points (M) and orthogonal tothe axis; then of obtaining information (R'f(U)) concerning the secondpoints (U); then of deducing from this by digital processing derivedinformation and then of combining this derived information concerninggroups of second points (U) belonging to the same meridian planes so asto deduce from the intermediate information (Q (ρ, B)) concerning thethird points (B); and finally to combine the intermediate information(Q(ρ, B)) concerning groups of third pints (V) belonging to the sameparallel planes in order to deduce from this the parameter (f(M))concerning the first points (M).